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Differentiation

Course Material

Gyeongsang National University

Dept. of Information & Communication Engineering

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Differentiation

• Differentiation (微分)

– Mathematical technique used to analyze changes in function – Analyze how rapidly a function changes at a particular point

• Voltage drop in capacitors, temperature change of chemical compounds, etc.

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Differentiation

• Average rate of change of function for interval

– Average rate of change for interval [t₁, t₂]

Change of function Change of interval

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Differentiation

• Average rate of change of function for interval

– Where should we move t₂ to obtain the instantaneous rate of change (slope of tangent) at point A?

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Limit of Function

• Limit of Function

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Limit of Function

• Example) Derive the limit at each point for 𝑦(𝑥)

𝑦 𝑥 = ቐ 1 − 𝑥, 𝑥 < 0 3, 𝑥 = 0 𝑥 + 1, 𝑥 > 0 (𝑎) lim 𝑥→3𝑦 (𝑏) lim𝑥→−1𝑦 (𝑐) lim𝑥→0𝑦 If lim

𝑥→𝑎𝑦(𝑥) ≠ 𝑦(𝑎), the function 𝑦(𝑥) is discontinuous at point

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Limit of Function

• Example) Answer the question for the following function 𝑦(𝑥).

𝑦 𝑥 = ቐ 0, 𝑥 < 0 𝑥, 0 < 𝑥 ≤ 2 𝑥 − 2, 𝑥 > 2 (𝑎) lim 𝑥→2+𝑦 (𝑏) lim𝑥→2−𝑦

At point 𝑥 = 𝑎, the limit does not exist when the left-side limit and the right-side limit are different.

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• Limit of Function

– At point 𝑥 = 𝑎, the limit exists when the left-side limit and the right-side limit are equal.

• Continuity of Function

– If lim

𝑥→𝑎 𝑦(𝑥) ≠ 𝑦(𝑎), the function 𝑦(𝑥) is discontinuous at point 𝑥 = 𝑎.

Limit and Continuity of Function

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Rate of Change at Point

• Find the average rate of change in 𝑦 𝑥 = 3𝑥

2

_{+ 2}

for the following intervals

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Rate of Change at Point

• For the interval [3, 3 + 𝛿𝑥], when 𝛿𝑥 approaches to

0, find the rate of change in 𝑦 𝑥 = 3𝑥

2

+ 2 at 𝑥 = 3.

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Rate of Change at Point

• Rate of Change at Point 𝑥

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• Derivative at arbitrary point 𝑥

• Derivative at particular point 𝑥

₀

Derivative

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Existence of Derivative

• The point that the derivative does not exist

– Discontinuous point

– Cusp or corner (point that the instantaneous rates of change from the left-side and right-side are different)

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Existence of Derivative

• Example) For the following functions, find the point that

does not exist the derivative

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Linearity Rules

• Differentiation is a linear operator

– Differentiation of the sum or difference of functions

– Differentiation of a function multiplied by a constant

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Product Rule

• Product rule

– When 𝑦 𝑥 = 𝑢 𝑥 𝑣(𝑥), derivative of 𝑦 𝑥 is given by

Alternative ∶ 𝑑𝑦 𝑑𝑥 = 𝑑𝑢 𝑑𝑥 𝑣 + 𝑢 𝑑𝑣 𝑑𝑥 – Example) a) 𝑦 = 𝑥 sin 𝑥 b) 𝑦 = 𝑡2𝑒𝑡

– Damped sinusoidal signal 𝑓(𝑡) = 𝑒−0.1𝑡 cos 𝑡

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Quotient Rule

• Quotient rule

– When 𝑦 𝑥 = 𝑢(𝑥) 𝑣(𝑥) , derivative of 𝑦 𝑥 is given by – Example) a) 𝑦 = sin 𝑥 𝑥 b) 𝑦 = 𝑡2 2𝑡+1 𝑦′ 𝑥 = 𝑣 𝑥 𝑢 ′ _{𝑥 − 𝑢 𝑥 𝑣′ 𝑥} 𝑣2 𝑥

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Chain Rule

• Chain rule

– When 𝑦 = 𝑦(𝑧), 𝑧 = 𝑧 𝑥 , 𝑦 can be represented by a function of 𝑥 and 𝑑𝑦 𝑑𝑥 is given by – Example) a) 𝑦 = 𝑧6 , 𝑧 = 𝑥2 + 1 b) 𝑦 = ln(3𝑥2 + 5𝑥 + 7) c) when 𝑦 = ln 𝑓(𝑥), 𝑑𝑦 𝑑𝑥 = 𝑓′(𝑥) 𝑓(𝑥)

𝑑𝑦

𝑑𝑥

=

𝑑𝑦

𝑑𝑧

×

𝑑𝑧

𝑑𝑥

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Parametric Differentiation

• Parametric differentiation

– When 𝑦 and 𝑥 are given by the functions of 𝑡 as 𝑦 = 𝑦(𝑡), 𝑥 = 𝑥 𝑡

– We call 𝑡 parameter. Useful when 𝑡 is hard to represented as a function of 𝑥 by removing 𝑡

– Example)

a) When 𝑦 = (1 + 𝑡)2 and 𝑥 = 2𝑡, find 𝑑𝑦 𝑑𝑥 b) When 𝑦 = 𝑒𝑡 + 𝑡 and 𝑥 = 𝑡2 + 1, find 𝑑𝑦

𝑑𝑥

c) When 𝑥 = sin 𝑡 + cos 𝑡 and 𝑦 = 𝑡2 − 𝑡 + 1, find 𝑑𝑦

𝑑𝑥 𝑡 = 0

𝑑𝑦

𝑑𝑥

=

𝑑𝑦

𝑑𝑡

×

𝑑𝑡

𝑑𝑥

=

𝑑𝑦

𝑑𝑡

/

𝑑𝑥

𝑑𝑡

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Differentiation of Implicit Form

• Differentiation of implicit Form

– When 𝑥3 _{+ 𝑦 = 1 + 𝑦}3_{, what is} 𝑑𝑦 𝑑𝑥? – Hard to represent as 𝑦 = 𝑓 𝑥

→ 𝑦 is represented in an implicit form of 𝑥 – Differentiation of both sides with 𝑥

– Example)

a) When 𝑥3 + 𝑦 = 1 + 𝑦3, find 𝑑𝑦 𝑑𝑥 b) When ln 𝑦 = 𝑦 − 𝑥2, find 𝑑𝑦

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Logarithmic Differentiation

• Logarithmic Differentiation

– Differential on products of complex functions can be easily obtained using logarithmic differentiation

– Differentiation by applying the natural logs (ln) – Example)

a) When 𝑦 = 𝑥3(1 + 𝑥)9𝑒6𝑥, find 𝑑𝑦 𝑑𝑥? b) When 𝑦 = 1 + 𝑡2sin2𝑡, find 𝑑𝑦

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Higher Order Derivatives

• Higher order derivatives

– We call 𝑑𝑦

𝑑𝑥 as the first derivative of 𝑦(𝑥) – We call 𝑑

𝑑𝑥 𝑑𝑦 𝑑𝑥 =

𝑑2𝑦

𝑑𝑥2 as the second derivative of 𝑦(𝑥) (can be represented by 𝑦′′)

– We call 𝑑𝑛𝑦

𝑑𝑥𝑛 as 𝑛-th order derivative of of 𝑦(𝑥)

– Example)

a) When 𝑦 𝑥 = 3𝑥2 + 8𝑥 + 9, find 𝑦′ and 𝑦′′ b) When 𝑦 𝑡 = 2 sin 3𝑡, find 𝑦′ and 𝑦′′

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Higher Order Derivatives

• Shape of the function according to the first and second

derivatives

(a) y is convex up (y > 0, y  > 0); (b) y is concave up (y> 0, y  < 0); (c) y is concave down (y < 0, y  < 0); (d) y is convex up (y < 0, y  > 0).

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Local Maximum and Local Minimum

• Local maximum and local minimum

– Local maximum – the point at which the function stops to increase and begins to decrease.

– Local minimum – the point at which the function stops to decrease and begins to increase.

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